(Short answer at the bottom.)

Russia conquered Finland from Sweden during the Napoleonic wars (1808-1809). Finland had been an integral part and a province of Sweden proper, but in the Russian empire Finland became a grand duchy, with the Russian emperor as the grand duke. Emperor Alexander I, who was at this point more liberal than later in his reign (and also worried about the loyalty of the Finnish subjects in context of the potentially upcoming war with France), promised to honour the ancient Finnish (ie. Swedish) "fundamental laws", traditional religion of the land (lutherian instead of Russian orthodoxy) and the freedom of Finnish peasants (Russian peasants were serfs, not free) in exhange of the loyality from the Finnish subjects. A diet was convened, where Alexander gave these promises and the four estates of Finland swore loyalty to him.

For the next few decades Finland had it's own government, laws, own language of administration (Swedish) and religion. Finland had its own minister in the Russian cabinet, and all the laws and imperial decisions were passed by the emperor with the minister for Finland instead of the Russian ministers. Finland also had its own army that was part of the Russian army, but separate from it. After the middle of 1800s Finland even started to get more autonomy: Finland gained its own national bank (a decade before the Russian national bank), its own currency (markka instead of the Russian rubble) and Finnish language was raised to parity with Swedish in the administration, the diet started to meet and pass laws (after the first diet Alexander I and his successor Nicholas I did not call the diet again for about 50 years). Emperor Alexander II was behind this larger autonomy, and his statues is still in the center of the main square in the Finnish capital.

At the end of the 1800s Russia started to russify Finland. Finland lost its army, part of its separate legistlation, separate postal system etc. Russian language was also introduced to the schools and administration. This led to a general strike during the Russian revolutions of 1905 and to a new, more democratic diet, but after the revolutionary period ended the russification began again. After the liberal revolution of 1917 the Finnish diet declared itself the highest authority apart from foreign and military affairs (they were reserved for the liberal provisional government of Russia), and after the communist revolution of Lenin the Finnish diet declare full independence from Russia 6.12.1917. Lenin and the communist government approved the declaration after brief negotiations with Finland in January 1918.

Russian and Finnish historians and politicians - both today and during the last two hundred years - have had different views on the status of Finland. Some have considered Finland an integral part of the Russian empire, some have felt that Finland was actually only in personal union with Russia, some have talked about a real union between the countries. The Russian imperial constitution of 1905 clearly states that Finland is a distinct area, but one that has been eternally bound to Russia and the Russian emperor shall always be the Finnish grand duke. There is actually a rather large amount of specialist literatury on the subject of the exact status of Finland, some of which is rather esoteric: some historians base their arguments for example on the spesific words and their meanings used by Alexander I on the first Finnish diet. Did he say "fundamental laws", "basic laws" or "constitution", and what exactly he meant with each word? What is the difference between a French word and a Russian word (Alexander I spoke both during the diet)? Some historians argue that the promises of Alexander I were not valid because he was forced to give them by the circumstanses, other historians say that only forced promises are valid at all (even I must say that this latter argument is completely stupid).

TL;DR for informed novices: Finland was an autonomic grand duchy in personal union with the Russian empire.

TL:DR for students of history: The status of Finland is a heavily discussed topic for historians, but Finland can broadly speaking be considered as an autonomic and distinct part of the Russian empire.

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Poland is not my specialty, but I understand that at the congress of Wienna in 1815 Poland was reformed as a partly self-governed Kingdom of Poland that was in personal union with the Russian empire. The brother of emperor Alexander I was named as a viceroy of Poland. Due to Polish revolts in 1830-1831 and 1864 Poland was stripped of its self-government and basically became a "normal" part of Russia. Like Finland, Poland was however also named as a distinct part of Russia in the imperial constitution of 1905.